Abstract

We investigate the existence of positive solutions of singular problem (−1)𝑚𝑥(2𝑚+1)=𝑓(𝑡,𝑥,…,𝑥(2𝑚)), 𝑥(0)=0, 𝑥(2𝑖−1)(0)=𝑥(2𝑖−1)(𝑇)=0, 1≤𝑖≤𝑚. Here, 𝑚≥1 and the Carathéodory function 𝑓(𝑡,𝑥0,…,𝑥2𝑚) may be singular in all its space variables 𝑥0,…,𝑥2𝑚. The results are proved by regularization and sequential techniques. In limit processes, the Vitali convergence theorem is used.